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Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium.

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Title: Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium.
Authors: Carrera, E.1 (AUTHOR) erasmo.carrera@polito.it
Source: Mechanics of Advanced Materials & Structures. 2025, Vol. 32 Issue 4, p672-684. 13p.
Subjects: Diophantine equations, Surface plates, Differential equations, Newton-Raphson method, Cartesian coordinates
Abstract: This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method's effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach. [ABSTRACT FROM AUTHOR]
Copyright of Mechanics of Advanced Materials & Structures is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium.
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  Data: <searchLink fieldCode="AR" term="%22Carrera%2C+E%2E%22">Carrera, E.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> erasmo.carrera@polito.it</i>
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  Data: <searchLink fieldCode="JN" term="%22Mechanics+of+Advanced+Materials+%26+Structures%22">Mechanics of Advanced Materials & Structures</searchLink>. 2025, Vol. 32 Issue 4, p672-684. 13p.
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  Data: <searchLink fieldCode="DE" term="%22Diophantine+equations%22">Diophantine equations</searchLink><br /><searchLink fieldCode="DE" term="%22Surface+plates%22">Surface plates</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+equations%22">Differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Newton-Raphson+method%22">Newton-Raphson method</searchLink><br /><searchLink fieldCode="DE" term="%22Cartesian+coordinates%22">Cartesian coordinates</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method's effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mechanics of Advanced Materials & Structures is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1080/15376494.2024.2353897
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      – Code: eng
        Text: English
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        PageCount: 13
        StartPage: 672
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      – SubjectFull: Diophantine equations
        Type: general
      – SubjectFull: Surface plates
        Type: general
      – SubjectFull: Differential equations
        Type: general
      – SubjectFull: Newton-Raphson method
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      – SubjectFull: Cartesian coordinates
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              Text: 2025
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